If $f(x) + 2f(1-x) = x^2 + 2 \; \forall x \in R $  , then f(x) is given as

Problem: If $f(x) + 2f(1-x) = x^2 + 2 \;  \forall x \in R $  , then f(x) is given as

(A) $\frac{(x-2)^2}{3} $

(B) x2 – 2

(C) 1

(D) None of these

Ans: (A)

Sol. By replacing x with (1 – x) in the given expression

f(1 – x) + 2 f(1 – 1 + x)

= (1 – x)2 + 2

⇒ f(1 – x) + 2 f(x)

= (1 – x)2 + 2

Now f(x) + 2 f(1 – x) – 2(f(1 – x) + 2f(x))

= x2 + 2 – 2((1 – x)2 + 2)

⇒  -3 f(x) = x2 + 2 – 2(3 – 2x + x2)

⇒  3 f(x) = x2 – 4x + 4

⇒  $ f(x) = \frac{(x-2)^2}{3} $